Origin of Granular Capillarity Revealed by Particle-Based Simulations

When a thin tube is dipped into water, the water will ascend to a certain height, against the action of gravity. While this effect, termed capillarity, is well known, recent experiments have shown that agitated granular matter reveals a similar behavior. Namely, when a vertical tube is inserted into a container filled with granular material and is then set into vertical vibration, the particles rise up along the tube. In the present Letter, we investigate the effect of granular capillarity by means of numerical simulations and show that the effect is caused by convection of the granular material in the container. Moreover, we identify two regimes of behavior for the capillary height $H_c^{\infty }$ depending on the tube-to-particle-diameter ratio, $D/d$. For large $D/d$, a scaling of $H_c^{\infty }$ with the inverse of the tube diameter, which is reminiscent of liquids, is observed. However, when $D/d$ decreases down to values smaller than a few particle sizes, a uniquely granular behavior is observed where $H_c^{\infty }$ increases linearly with the tube diameter.

Figure 1

Granular capillarity effect. When a narrow tube is dipped into granular material and vibrated vertically, the granular material rises inside the tube to reach a terminal vertical level. The snapshots (a)–(k) show the system at time, $t$, in units of the tube’s oscillation period, $T=1/f\approx 83\mathrm{ms}$. System parameters are provided in Sec. I of the Supplemental Material [13]. The tube-to-particle-diameter ratio is $D/d=13.33$. Particles are color coded based on their initial positions at time $t=0$, panel (a).

Figure 2

Capillary height, $H_c$, as a function of time. $z_b$ denotes the vertical position of the tube’s lower end, see Eq. (1).

Figure 3

Steady state capillary height, $H_{\mathrm{c}}^{\infty }$, as a function of $D/d$. Two different regimes are identified. For $D/d<10$, we obtain a linear increase of $H_c^{\infty }$ with $D/d$, while for larger $D/d$ the scaling with $d/D$ describes well the simulation data. Lines show best fits to the simulation data for both regimes.

Figure 4

Average vertical velocity $⟨v_z⟩$ of the particles in the tube as a function of the squared nondimensional radial position, $(r/R_{\mathrm{tube}}{)}^2$. Circles denote results for the ascending phase ($t\lesssim 20T$), while squares correspond to the steady-state phase ($t\gtrsim 50T$). The dotted line indicates $⟨v_z⟩=0$.

Figure 5

Number fraction $P_{\mathrm{out}}$ of particles that were at the outer volume of the tube at $t=0$. $P_{\mathrm{out}}$ is shown as a function of the vertical position $z$ for different values of the radial position $r$. (a) and (b) refer to simulation results at $t=10T$ and $t=75T$, which correspond to snapshots in the ascending and steady-state regimes, respectively.